Finite Group Databasebeta

← #10705794

Subgroups — #10705794

Order 768.

590 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76780C211yesmaximal, characteristic
2#68952C211yesmaximal, characteristic
3#68103C211yesmaximal, characteristic
4C3×#1351C211yesmaximal, characteristic, complemented
5#67646C211yesmaximal, characteristic, complemented
6#67490C211yesmaximal, characteristic, complemented
7#67449C211yesmaximal, characteristic, complemented
8#11785—31nomaximal, Sylow
9C22×D3×Q8C411yescharacteristic, complemented
10C2×#67⋊C4C2211yescharacteristic
11C2×#878—21no—
12C3×#393C2211yescharacteristic
13#5230C411yescharacteristic, complemented
14#4956—21no—
15C2×#806—21no—
16#4947C2211yescharacteristic
17Dic3×C22⋊C4—21no—
18#4719—21no—
19#4668—21no—
20#4667C411yescharacteristic, complemented
21#4656C411yescharacteristic, complemented
22#4646—21no—
23C3×#342C2211yescharacteristic
24C3×#323C2211yescharacteristic
25#4526C2211yescharacteristic
26#4525C2211yescharacteristic
27#4468—21no—
28#2409—31no—
29C4×#99—31no—
30#1466—31no—
31#1351—31no—
32#1351—31no—
33#1351—31no—
34#1351D311yescharacteristic, complemented
35C22×C3×Q8C2×C411yescharacteristic
36C22×#67D411yescharacteristic
37C23×Dic3D411yescharacteristic
38C2×D3×Q8—44no—
39C2×D3×Q8—41no—
40C2×D3×Q8—41no—
41C22×C4×D3—21no—
42C22×C4×D3C2×C411yescharacteristic
43C22×Dic6—21no—
44C22×Dic6C2×C411yescharacteristic
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4C2311yescharacteristic
48C2×#217—21no—
49C2×#217—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4D411yescharacteristic
52C2×C4⋊Dic3—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4C2×C411yescharacteristic
56C2×C4×Dic3—41no—
57C2×C4×Dic3D411yescharacteristic
58#67⋊C4—41no—
59#67⋊C4—41no—
60#878—44no—
61C3×C22⋊C8—21no—
62C3×#95—21no—
63C3×#95C2×C411yescharacteristic
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—41no—
68C3⋊#95C2×C411yescharacteristic
69#830—21no—
70#815—41no—
71Dic3⋊C8—41no—
72#806—24no—
73C23×Q8—31no—
74C2×C22⋊Q8—61no—
75C2×C4×D4—31no—
76#432—61no—
77C2×#99—61no—
78#395—31no—
79#395—31no—
80#393—31no—
81#393C2×D311yescharacteristic
82#391—61no—
83#386—61no—
84#385—61no—
85#380—31no—
86C4×C22⋊C4—61no—
87#342—31no—
88#342C2×D311yescharacteristic
89#323—31no—
90#323—61no—
91#323C2×D311yescharacteristic
92C24×C3C2×D411yescharacteristic
93C23×D3C22⋊C411yescharacteristic
94C2×C3×Q8—42no—
95C2×C3×Q8—41no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—41no—
99C22×C3×C4C22×C411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C2×#67—21no—
102C2×#67—21no—
103C2×#67—22no—
104C2×#67—41no—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—42no—
110C22×Dic3—41no—
111C22×Dic3C22⋊C411yescharacteristic
112C22×Dic3C22⋊C411yescharacteristic
113C22×Dic3C22⋊C411yescharacteristic
114D3×Q8—44no—
115D3×Q8—44no—
116D3×Q8—44no—
117D3×Q8—44no—
118C2×C4×D3—21no—
119C2×C4×D3—21no—
120C2×C4×D3—42no—
121C2×C4×D3—42no—
122C2×C4×D3—42no—
123C2×C4×D3—42no—
124C2×Dic6—24no—
125C2×Dic6—42no—
126C2×Dic6—41no—
127C2×Dic6—41no—
128C2×Dic6—41no—
129C2×Dic6—42no—
130C2×C3×C8—41no—
131C3×C22⋊C4—22no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—42no—
134C3×C22⋊C4—41no—
135#217—22no—
136#217—42no—
137#217—41no—
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—42no—
140C4⋊Dic3—42no—
141Dic3⋊C4—41no—
142Dic3⋊C4—42no—
143Dic3⋊C4—42no—
144C4×Dic3—21no—
145C4×Dic3—21no—
146C4×Dic3—82no—
147C2×C3⋊C8—41no—
148C22×Q8—31no—
149C22×Q8—61no—
150C22×Q8—121no—
151C22×Q8—121no—
152C22×Q8—124no—
153C22×Q8C4×D311yescharacteristic
154C22×D4—31no—
155C23×C4—31no—
156C23×C4—31no—
157C23×C4—61no—
158C22⋊Q8—124no—
159C4×D4—121no—
160C4×D4—121no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—61no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—31no—
168C2×C22⋊C4—61no—
169C2×C22⋊C4—61no—
170C2×C22⋊C4—61no—
171C2×C22⋊C4C22×D311yescharacteristic
172C2×C42—31no—
173C2×C42—121no—
174C4⋊C8—121no—
175#99—64no—
176C22⋊C8—21no—
177C22⋊C8—61no—
178#97—121no—
179#95—21no—
180#95—31no—
181#95—61no—
182#95—61no—
183#95—61no—
184#95—121no—
185#95C4×D311yescharacteristic
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3C2×C22⋊C411yescharacteristic
191C23×C3#13711yescharacteristic
192C23×C3#13711yescharacteristic
193C22×D3—21no—
194C22×D3—21no—
195C22×D3—41no—
196C22×D3—41no—
197C22×D3#10412yes—
198C3×Q8—42no—
199C3×Q8—42no—
200C2×C3×C4—22no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—42no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—81no—
212C2×C3×C4—81no—
213#67—42no—
214#67—42no—
215C2×Dic3—22no—
216C2×Dic3—22no—
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—21no—
220C2×Dic3—21no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—42no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—42no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—81no—
230C2×Dic3—81no—
231C2×Dic3—81no—
232C2×Dic3#9911yescharacteristic
233C2×Dic3#9911yescharacteristic
234C4×D3—42no—
235C4×D3—48no—
236C4×D3—42no—
237Dic6—44no—
238Dic6—42no—
239Dic6—44no—
240Dic6—42no—
241C3×C8—81no—
242C3⋊C8—81no—
243C24—31no—
244C24D3×D411yescharacteristic
245C2×Q8—42no—
246C2×Q8—41no—
247C2×Q8—64no—
248C2×Q8—121no—
249C2×Q8—121no—
250C2×Q8—121no—
251C2×Q8—122no—
252C2×Q8—122no—
253C2×Q8—124no—
254C2×Q8—124no—
255C2×Q8—124no—
256C2×Q8—124no—
257C2×D4—62no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—121no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—31no—
264C22×C4—31no—
265C22×C4—31no—
266C22×C4—41no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—122no—
274C22×C4—122no—
275C22×C4—122no—
276C22×C4—122no—
277C22×C4—122no—
278C22×C4—121no—
279C22×C4C2×C4×D311yescharacteristic
280C22×C4D3×D411yescharacteristic
281C2×C8—41no—
282C2×C8—121no—
283C4⋊C4—121no—
284C4⋊C4—122no—
285C4⋊C4—122no—
286C4⋊C4—122no—
287C22⋊C4—22no—
288C22⋊C4—41no—
289C22⋊C4—42no—
290C22⋊C4—41no—
291C22⋊C4—62no—
292C22⋊C4—121no—
293C22⋊C4—122no—
294C22⋊C4—122no—
295C22⋊C4—121no—
296C42—61no—
297C42—61no—
298C42—242no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—81no—
308C22×C3C2×#9911yescharacteristic
309C22×C3#41911yescharacteristic
310C22×C3C2×#10411yescharacteristic
311C2×D3—22no—
312C2×D3—42no—
313C2×D3—41no—
314C2×D3—41no—
315C2×D3—42no—
316C3×C4—42no—
317C3×C4—41no—
318C3×C4—41no—
319C3×C4—81no—
320C3×C4—82no—
321Dic3—21no—
322Dic3—42no—
323Dic3—41no—
324Dic3—41no—
325Dic3—81no—
326Dic3—82no—
327Dic3AΣL(1,𝔽16)12yes—
328C23—21no—
329C23—21no—
330C23—32no—
331C23—41no—
332C23—41no—
333C23—61no—
334C23—61no—
335C23—121no—
336C23—121no—
337C23D3×C22⋊C411yescharacteristic
338C23#91511yescharacteristic
339C23#91811yescharacteristic
340Q8—42no—
341Q8—42no—
342Q8—124no—
343Q8—124no—
344Q8—122no—
345Q8—122no—
346D4—122no—
347D4—122no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—31no—
352C2×C4—31no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—62no—
362C2×C4—62no—
363C2×C4—61no—
364C2×C4—61no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—122no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—128no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—122no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—241no—
381C2×C4—241no—
382C2×C4—241no—
383C8—81no—
384C8—241no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—41no—
388C2×C3—41no—
389C2×C3#150811yescharacteristic
390C2×C3C2×AΣL(1,𝔽16)11yescharacteristic
391C2×C3#212911yescharacteristic
392D3—42no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—62no—
402C22—81no—
403C22—122no—
404C22—122no—
405C22—121no—
406C22—121no—
407C22D3×#9911yescharacteristic
408C22#480211yescharacteristic
409C22D3×#10411yescharacteristic
410C4—32no—
411C4—42no—
412C4—41no—
413C4—41no—
414C4—61no—
415C4—81no—
416C4—82no—
417C4—122no—
418C4—121no—
419C4—121no—
420C4—241no—
421C4—242no—
422C3#1178511yescharacteristic, complemented, Sylow
423C2—21no—
424C2—21no—
425C2—41no—
426C2—41no—
427C2—122no—
428C2#6833511yescharacteristic
429C2D3×AΣL(1,𝔽16)11yescharacteristic
430C2#6915911yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1351C2
socleC22×C3C2×#99
prime radicalC22×#67D4
cosocleC2×C3×C22⋊C4C23
hypercenter#1351D3
hypocenterC3#11785
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1351

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C3×#1351C2semidirect
5. #67646C2semidirect
6. #67490C2semidirect
7. #67449C2semidirect
9. C22×D3×Q8C4semidirect
13. #5230C4semidirect
20. #4667C4semidirect
21. #4656C4semidirect
34. #1351D3semidirect
422. C3#11785semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#767802
#689522
#681032
C3×#13512
#676462
#674902
#674492
#117853